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Journal of Financial Markets xxx (2018) 1e23

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Journal of Financial Markets

journal homepage: www.elsevier .com/locate/finmar

Liquidity might come at cost: The role of heterogeneouspreferences*

Shmuel Hauser a, Haim Kedar-Levy b, *

a Israel Securities Authority, Ono Academic College, and the Guilford Glazer Faculty of Business and Management, Ben-Gurion Universityof the Negev, Israelb The Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev, Israel

a r t i c l e i n f o

Article history:Received 12 June 2017Received in revised form 11 March 2018Accepted 19 March 2018Available online xxx

JEL Codes:C61D53E44G11G12

Keywords:HeterogeneityDiscount rate riskTurnoverLiquiditySharpe ratio

* We thank Yakov Amihud, Gideon Saar (the Ediindebted also to Zvi Afik, Doron Avramov, Scott CedeWiener. We thank participants of the Midwest Econthe Finance Department Research Seminar at the Hexpressed in this article do not necessarily reflect t* Corresponding author. P.O.B. 653, Beer Sheva 84

E-mail addresses: [email protected] (S. Hau

https://doi.org/10.1016/j.finmar.2018.03.0011386-4181/© 2018 Published by Elsevier B.V.

Please cite this article in press as: Hauser, SJournal of Financial Markets (2018), https:

a b s t r a c t

Asset-pricing models with volume are challenged by the high turnover-rates in real stockmarkets. We develop an asset-pricing framework with heterogeneous risk preferences andshow that liquidity and turnover increase with heterogeneity to a maximum, and thendecline. With U.S. parameters, turnover exceeds 55%. Liquidity is costly since it facilitates alarge share redistribution across agents, causing changes in average risk aversion, whichincreases Sharpe ratio variability, and hence stock return volatility. Illiquidity and its riskare minimized at moderate heterogeneity levels, highlighting an "optimal" heterogeneitylevel, yet, there is no "optimal" combination between liquidity level and Sharpe ratiovariability.

© 2018 Published by Elsevier B.V.

Frictionless asset pricing models such as the CAPM (Sharpe, 1964; Lintner, 1965) or ICAPM (Merton, 1971, 1973) are notstructured to account for volume or liquidity as they effectively assume perfectly elastic supply and demand for shares.However, real securities markets reveal prices by clearing bilateral supply and demand, and their performance is evaluated,inter alia, based on their liquidity (Amihud and Mendelson, 1987), as well as their Sharpe ratio volatility. The absence ofvolume in frictionless models motivated researchers to develop alternative models that incorporate volume in various ways,aiming to corroborate actual trading activity (see a brief literature review in Appendix 1). While some models assumeexogenous perturbations, others assume heterogeneous preferences, yet turnover is an order of magnitude smaller than

tor), Fernando Zapatero, and two anonymous referees for valuable suggestions and insights. We arerburg, David Feldman, Dan Galai, Arieh Gavious, Eugene Kandel, Michel Robe, Itzhak Venezia, and Zviomic Theory Conference, Lansing, MI, the Far Eastern Meeting of the Econometric Society, Beijing, andebrew University, Jerusalem. We assume full responsibility for any remaining errors. The opinions

he position of the Israel Securities Authority.105, Israel.ser), [email protected] (H. Kedar-Levy).

., Kedar-Levy, H., Liquidity might come at cost: The role of heterogeneous preferences,//doi.org/10.1016/j.finmar.2018.03.001

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S. Hauser, H. Kedar-Levy / Journal of Financial Markets xxx (2018) 1e232

measured in real markets. Moreover, we are not aware of a formal model with a causal interaction between levels of liquidityand Sharpe ratio volatility.

This paper belongs to the heterogeneous preferences strand of the literature, as we configure heterogeneity in away that isdifferent from existing models. We find that realistic levels of turnover and equity premium (EP) volatility emerge atmoderate heterogeneity levels. Volume, turnover, and liquidity are negligible at extreme homogeneity or heterogeneity, butinterior maxima exist. Thus, more heterogeneity would not necessarily increase market activity and liquidity. Particularly, atmoderate heterogeneity, high turnover and liquidity imply a higher redistribution of shares, causing large changes in averagerelative risk aversion (RRA), which increases EP volatility and consequently stock return volatilities. The notion that higherliquidity causes higher stock return volatility reveals a cost of liquidity; put differently, the appetite for liquidity should berestrained.

More specifically, we develop a frictionless, continuous timemodel with a riskless bond and a single risky asset that followgeometric Brownianmotions. We derive optimum portfolio rebalancing trades for two representative investors having powerutility functions. Investors differ by their level of the RRA parameter. In essence, we show that RRA parameters that bracketthemarket price of (variance) risk, l, imply that investors’ optimal portfolio rebalancing trades yield either an upward-sloping“supply” or a downward-sloping “demand” function for shares. For RRA<l, a “supply” function emerges in terms of units ofshares: buying (selling) when the stock price increases (declines), implying a positive feedback, a.k.a. “trend-chasing"strategy. This investor is denoted as “type-T,” with dT≡RRAT . Conversely, the “demand” function emerges when l<RRA,implying a “contrarian” trading strategy that is implemented by type-C investors with dC≡RRAC . These are “generalizedpreferences” as each type may be a representative agent of its group and both groups exist in any market because the har-monic mean of RRAs determines l.

We derive closed-form expressions of conditional traded quantities for each of the two investor types, and analyzebilateral trade volume. Since average RRA becomes an endogenous state variable, we use the martingale representationapproach to identify the relevant hedging component and study its impact. The core of the model is a formal analysis of theinteractions between preferences and Sharpe ratio volatility, turnover, volume, liquidity, and liquidity risk. We simulate abenchmark financial market based on NYSE post World War II statistics, and replicate it across 16 market states, from themost homogeneous (the dispersion of RRAs about l is minimal) to the most heterogeneous, where dT ¼ 1:01 and dC ¼ 10,with averages ranging from 3.39 to 3.85.

Our major findings are: 1) Trading volume is miniscule in the homogeneous case, increases to a maximum at moderateheterogeneity levels, and declines as heterogeneity keeps increasing. Minimal volume at the homogeneous case stems fromthe fact that when RRAs near l, investors' optimal asset allocation rules approximate buy-and-hold strategies. However,volume declines at high heterogeneity levels due to the extreme difference in RRAs: with an extremely low dT , type-T in-vestors allocate much of their wealth to equity, while an extremely high dC implies that type-C investors hold mostly bonds.As a result, type-C investors hardly trade shares, thus limiting traded volume. 2) Turnover rate increases with heterogeneity toamaximum, and then declines. Atmoderate heterogeneity levels (RRAs ~1.5 to ~6.5), themodel yields turnover of about 56% ifportfolio rebalancing frequency is daily, and standard deviation of the risky asset's return is 20%. It may exceed 180% if tradingfrequency is 4/day, and standard deviation is 35%, representing turbulent episodes. These magnitudes of market activityconform to empirical evidence in advanced financial markets. 3) Market illiquidity, defined as the absolute rate of returndivided by volume (Amihud's, 2002 ILLIQ), is high at both extremes, but a minimum illiquidity level exists at moderateheterogeneity (RRAs ~2.0 to ~5.5). 4) Illiquidity risk declines to a minimum and then increases with heterogeneity. Togetherwith the previous finding, a V-shape plot emerges of illiquidity versus its standard deviation (i.e., at some moderate het-erogeneity level, the combination of illiquidity and its risk is minimized). 5) Sharpe ratio varies in response to changes inaverage RRA; this variation is minimal at both extremes of heterogeneity, yet at moderate heterogeneity levels an increase inaverage RRA from 3.4 to 4.1 increases Sharpe ratio by about 18%, to 0.53. 6) While one might presume that more liquidity ispreferable to less liquidity, we find that since it allows greater share redistribution, high liquidity increases the magnitude ofchanges in average RRA, which induces a higher Sharpe ratio and hence stock price volatility. There is no single, “optimal”combination between the level of liquidity and Sharpe ratio variability.

In Section 1, we discuss the economic setting and optimal portfolio rules. In Section 2, we derive agents' motivation totrade, and in Section 3 solve for trading volume, together with comparative static analyses. In Section 4, we describe ourcalibration and simulation procedures, while in Section 5 we examine predictions pertaining to Sharpe ratio level andvariability in the cross-section of heterogeneity. In Section 6, we study cross-sectional predictions for turnover, liquidity, andliquidity risk. We conclude in Section 7.

1. The economic setting

Assume a single bond and a single risky asset trade in frictionless financial markets under information symmetry. Theriskless bond has a price Bt and it yields a constant rate of return r, following the process:

dBt=Bt ¼ rdt: (1)

Please cite this article in press as: Hauser, S., Kedar-Levy, H., Liquidity might come at cost: The role of heterogeneous preferences,Journal of Financial Markets (2018), https://doi.org/10.1016/j.finmar.2018.03.001

S. Hauser, H. Kedar-Levy / Journal of Financial Markets xxx (2018) 1e23 3

The risky asset is an open-ended mutual fund that holds the market portfolio of stocks. It is a claim on the aggregatedividend, D, generated by an exogenous process:

dDt=Dt ¼ mDdt þ sDdzt ; (2)

where both mD and sD are given constants. The stock has an exogenously-given standard deviation s and its price, P, evolves by
the process:
dPt=Pt ¼ dSt=St þ Dtdt ¼ mtðpT ; tÞdt þ sdzt ; (3)

where S is the ex-dividend price, and mtðpT ; tÞ is an instantaneous total expected rate of return (to be determined in equi-
librium). The model features two investor types, T and C, both having power utility functions. They differ only with respect totheir relative risk aversion (RRA) parameter in a particular, yet generic way as detailed in the next section. Let pT ;t be investortype-T's share out of all outstanding shares at t. The remaining shares are held by investor type-C, i.e., pC;t ¼ 1� pT ;t ct. Inthe presence of heterogeneity, pT ;t is an endogenous state variable as share redistribution changes the weighted average RRAin the market. Investor type indexing is omitted throughout most of this section, to simplify notation. Assume that pT followsthe process:
dpT ¼ mpðpT ; tÞdt þ spdzt ; (4)

where sp is an exogenous standard deviation that multiplies the Brownian motion dzt , and dztdzt ¼ fdt. Let ct be the
consumption rate per period and let at be the fraction of individual wealth, Wt , invested in the risky asset at t. Thus,
dW ¼ Wt

�atdPtPt

þ ð1� atÞdBtBt

�� ctdt

¼ ½atðmtðpT ; tÞ � rÞ þ r�Wtdt � ctdt þ atWtsdzt :

(5)

Each investor/consumer solves the following:

Maxcs;as

Et

264ZTt

Uðcs; sÞdsþ H�WT ; T

�375; (6)

subject to the budget constraint (5), where H is an increasing and concave utility function of bequest and T is the terminaldate.

Using martingale representation, assume that the Brownian motion of the state variable pT , denoted in (4) as dzt , is linearwith dzt , therefore changes in the investment opportunity set are perfectly correlated with the stock price, and the market isdynamically complete. Since the stock can be used to hedge against changes in the investment opportunity set, we can invokethe hedging concept of Black and Scholes. Assuming further the absence of arbitrage, there exists a stochastic discount factorthat evolves by the following process:

dM=M ¼ �rdt �QðtÞdz; (7)

where Q is the Sharpe ratio, satisfying:

ðmtðpT Þ � rÞ=s ¼ Qt : (8)

The technical part of the derivation procedure is detailed in Appendix 2. Important for our purposes here, the solutionyields the following optimum asset allocation (equation (A14) in Appendix 2):

at ¼ �MWM

Wm� rs2

þWpTspWs

: (9)

Now recall that both agent types, indexed by K ¼ fT;Cg, have power utility functions of the following form:

UKðcÞ ¼ e�rtcgK =gK : (10)

Utility function (10) features a similar rate of time preference r but different individual RRAs. Note that by analogy to theoptimality conditions of stochastic dynamic programing, where the state variable is perfectly correlated with stock returns,MWM ¼ JW=JWW and WpT ¼ � JWpT

=JWW . We may replace in the first addend of (9) MWM ¼ WgK�1=ðgK � 1ÞWgK�2. Further,by the inverse optimummethod, we may “guess” a policy function and verify that it is consistent with the objective functionand the constraints. If it is, the assumed policy function is consistent with optimizing behavior. Therefore, we “guess” that:



mt�pT;t

� ¼ r þ s2

Jt¼ r þ s2

pT ;t

dTþ 1�pT;t

dC

; (11)

that is, the expected rate of return on the risky asset is the sum of the riskless rate and compensation for bearing risk. Risk in(11) is scaled by Jt, the weighted average risk tolerance in the market, in which dT≡1� gT and dC≡1� gC are the investors'RRAs. As shown below in (24), (11) indeed emerges as an optimal solution.

To find WpT one should express the functional relation between pT ;t and consumption through vc=vW and vc=vpT ;t . Asshown by Merton (1969, 1971), with zero bequest and power utility, consumption is linear in wealth,

c*K;t ¼aK

1� e�aKðT�tÞWK;t ; (12)

"ðm�rÞ2

#
where, aK≡1�dK
dK

r1�dK

� r� 2dKs2 . For the infinite horizon case and for mtðpT ;tÞ we have,

c*K;t ¼1� dKdK

"r

1� dK� r �

�mt�pT;t

�� r�2

2dKs2

#WK;t : (13)

Replacing (11) into (13) and simplifying we have,

c*K;t ¼1� dKdK

"r

1� dK� r � s2

J2t 2dK

#WK;t

¼ r� rð1� dKÞdK

WK;t �s2ð1� dKÞ�

pT ;t

dTþ 1� pT;t

dC

2

2d2K

WK;t :(14)

Therefore, the partial derivative vc=vpT ;t≡cpT can be solved in closed form as:

cpT ¼ s2ð1� dK ÞWK;t

d2K

264pT;t

�1dT

� 1dC

� 2dTdC

þ 1dC

�1dT

� 1dC

�pT;t

dTþ 1� pT ;t

dC

4

375

≡s2ð1� dKÞWK;t

d2Kht :

(15)

where ht is the term inside the square bracket in the first line of (15). Similarly, by (14), vc=vW≡cW in the hedging demand
addend of (9),
cW ¼ 1� dKdK

"rK

1� dK� r � s2

J2t 2dK

#: (16)

By replacing (15) and (16) into (9), we have a solution to a*K;t ,

a*K;t ¼mt�pT;t

�� rdKs2

þ htssp

dK

�rK

1�dK� r � s2

J2t 2dK

; dK >1: (17)

We focus on dK >1 (i.e., RRA higher than log utility). While the first addend in (17) is the familiar mean-variance demandfor the risky asset, it would be interesting to study the role that hedging demand plays in affecting a*K;t , and market activity.

We explore the marginal impact of the hedging demand component on volume and liquidity for different RRAs in Sections 4and 5.

Our key assumptions are as follows: A dividend process with constant parameters determines the stock price process,which features a constant standard deviation but state-dependent expected return. The state variable is relative share-holdings of our two investor types, who have different RRAs in a power utility function. Assuming symmetric information,each investor optimizes consumption and asset allocation. Thus, changes in relative shareholdings change aggregate RRA. We



use the martingale technique to solve the problem, assuming that the state variable is perfectly correlated with the priceprocess, with no arbitrage opportunities.

2. The motivation to trade

This section is devoted to deriving optimal intertemporal trade by each investor type, aiming to express volume andliquidity in closed form. Bilateral trading volume emerges in our setup through agent heterogeneity, measured by thedispersion of RRAs about the market price of (variance) risk. The formal derivation is based on (20), where two mutually-exclusive optimal rebalancing strategies imply conditional buy or sell orders for shares; conditional on the direction ofprice change (see Gavious and Kedar-Levy, 2013). These schedules, representing instantaneous intertemporal supply and

demand for shares, are derived by rewriting (17) in terms of quantities and prices. Let 4K;t≡htssp

rK1�dK

�r� s2

J2t 2dK

and lt≡mt�rs2 . Multiply

both sides of (17) byWK;t to get a*K;tWK;t ¼�ltþ4K ;t

dK

WK;t and represent holdings in terms of quantities and prices of shares and

bonds,

NK;tPt ¼�lt þ 4K;t

dK

�NK;tPt þ QK;tBt

�; (18)

where N denotes the quantity of shares, Q denotes the quantity of bonds, and P and B denote their prices, respectively. Note
that the quantities in (18) are post-rebalancing. At t þ dt, prior to portfolio rebalancing, period t quantities are multiplied byperiod t þ dt prices, and (18) takes the form:
NK;tþdtPHtþdt ¼

�ltþdt þ 4K;tþdt

dK

�NK;tP

Htþdt þ DK;tð1þ rdtÞ

�; (19)

where PH is a variable that obtains a continuum of hypothetical stock prices for which we construct the implied sup-
tþdt
plyydemand schedules; in equilibrium PHtþdt :¼ Ptþdt . It is advantageous to represent bond holdings at t, DK;t ¼ QK;tBt , as afunction of shareholdings. Solve for DK;t in (18) as follows:

DK;t ¼ SK;t��

lt þ 4K;t�

dK�� SK;t : (20)

Defining the number of shares held by investor K at t þ dt as the number of shares held at t plus an optimal addition(subtraction) through trade over dt, we may replace NK;tþdt with NK;t þ dNK;tþdt on the left-hand side of (19) and solve for the

functional relation between hypothetical prices, PHtþdt , and trade, dNK;tþdt . Replacing (20) into (19) and solving for PHtþdt, wehave1:

PHtþdt ¼SK;t

�1��

lt þ 4K;t�

dK�� 1

��ltþdt þ 4K;tþdt

�dK

�ð1þ rdtÞNK;t

�1� ��

ltþdt þ 4K;tþdt�

dK��þ dNK;tþdt

: (21)

Investor K's RRA determines whether the slope of (21) in the hypothetical pricemarginal trade plane is positive or negative(i.e., whether the investor would buy or sell given an increase in PHtþdt). A partial derivative of (21) yields

vPHtþdt

vdNK;tþdt¼ �

SK;thltþdtþ4K;tþdt

ltþ4K;t� �


dK

ið1þ rdtÞ�

NK;t�1� �


dK�þ dNK;tþdt

�2 : (22)

The derivative in (22) will be negative if the term in the square brackets of the numerator is positive. SK;t>0, i.e., a longposition in the stock is assumed throughout for both agents.

Proposition 1 formalizes investors’ separation into two mutually exclusive groups:

Proposition 1. (Optimal Intertemporal Trade): If the utility parameter 1< dT satisfies dT < ðltþdt þ 4T ;tþdtÞ, (21) obtains apositive slope, thus defining the investor as positive feedback (i.e., trend-chasing (type-T)). Alternatively, if dC > ðltþdt þ 4C;tþdtÞ, the

1 Equation (24) is comparable in essence to equation (3) in Johnson (2008), where changes in asset allocation are analyzed under the condition of valueneutrality.


Fig. 1. Implied marginal trade schedules.


slope in (24) is negative and the investor is defined as contrarian (type-C). Thus, the type-T investor is less risk-averse than a type-Cinvestor, and because those RRAs bracket the market price of risk, both types exist in equilibrium in any market.

Proof. See Appendix 3.RRA parameters distinguish between both schedules through the ranking 1< dT < ðltþdt þ 4K;tþdtÞ< dC , K ¼ fT ;Cg. These

schedules are not “supply” or “demand” functions for shares in the conventional meaning, since both supply and demand arepresent along each curve, depending on the sign of price changes. Thus, the term marginal trade schedules (MTS) appearsmore appropriate. This notion is presented graphically in Fig. 1.

Our investors' trades are independent of the other party, thus the period net demand need not be zero, as can be seen inFig. 1. To secure equilibrium, the number of outstanding shares is adjusted in accordance with excess demand or supply, as inMerton (1971, 1973) or Dumas (1989). Equilibrium market clearing implies NtþdtPtþdt ¼

PKNK;tþdtPtþdt (i.e., total outstanding

number of shares in the market equals the sum of individual demands).2 Thus, after portfolio rebalancing:

NtþdtPtþdt ¼X

K¼C;T

��ltþdt þ 4K;tþdt

�dK

��NK;tþdtPtþdt þ QK;tþdtBtþdt

�: (23)

The equilibrium expected stock return can be derived by using (18), (19), and (20). Applying the market clearing conditionNC;tþdt ¼ Ntþdt � NT ;tþdt , and replacing bond holdings with their equivalent magnitude in terms of stock holdings (as both aredetermined simultaneously),3 one obtains:

mt�pT;t

�� r ¼ s2

pT;t

dTþ 1�pT;t

dC

; (24)

which is similar to (11). Equation (24) highlights that because the less risk-averse investors buy (sell) shares when the priceincreases (declines), weighted average risk tolerance, Jt ¼ pT ;t=dT þ ð1� pT ;tÞ=dC , increases (declines) due to time variationin share allocation between the two investor types.4

3. The determinants of volume

These building blocks allow the exploration of the ways volume and liquidity vary with key parameters. Our first step isderiving closed-form expressions for optimal rebalancing of volume between t and t þ dt by each investor type, C and T

2 As noted, in some representative investor models (e.g., Merton, 1971, 1973; Lucas, 1978), total float is adjusted to equal aggregate demand for riskyassets to secure equilibrium. Yet, absent bilateral trade, volume in such models has little meaning and is not comparable to volume as measured in realfinancial markets. Similarly, the infinite volume that emerges in continuous-time models is not indicative of actual volume (see Wang, 1994).

3 To obtain (24) and (11), rewrite (18) as, where SK;t is stocks and DK;t is bonds. Solve for D: DK;t ¼ SK;t=ðdK=ðlt þ fK;tÞÞ� SK;t , and replace it in QK;tBtþh ¼DK;tð1þ rdtÞ. Divide by Pt and simplify.4 Note that the equity premium is inversely related to the price level, a property shared with other heterogeneous preferences models (e.g., Garleanu and

Panageas, 2015; Bhamra and Uppal, 2009, 2014).



(henceforth, “trade plans”). Trade plans differ from bilateral volume, as the latter is the minimum between the two tradeplans, in absolute terms.

3.1. Closed-form expressions for volume

Proposition 2. (The drivers of volume): Portfolio rebalancing volume by investor type-T is:

VOLðTtþdtÞ≡dNT;tþdt

¼ NtþdtPtþdtð1� bÞ � NT;tPtþdtða� bÞ � aWT;tþdtð1� aÞ � bWC;tþdtð1� bÞPtþdt

�ltþdtð1=dT � 1=dCÞ � 4C;tþdt

dC þ 4T ;tþdt

dT� (25)

and rebalancing volume by investor type-C is given by:

VOLðCtþdtÞ≡dNC;tþdt

¼ NtþdtPtþdtð1� aÞ � NC;tPtþdtðb� aÞ � aWT ;tþdtð1� aÞ � bWC;tþdtð1� bÞPtþdt

�ltþdtð1=dC � 1=dT Þ þ 4C;tþdt

dC � 4T;tþdt

dT� (26)

where a≡ltþdtþ4T;tþdt

dT, b≡ltþdtþ4C;tþdt

dC.

Proof. See Appendix 4Trade plans between t and t þ dt depend on investors’ asset allocations at the beginning of the period (NC;t ; NT ;t), on their

wealth, number of outstanding shares at t þ dt, on hedging demands, and most interestingly, on the level and dispersion ofboth RRAs about ltþdt . The hedging terms 4K;tþdt decline toward zero when trading volume declines due to the smallervariability in weighted-average RRA (WA-RRA). This occurs when RRAs are either near or far from ltþdt .

A brief review of the volume equations reveals that the closer dT is to dC , and both of them to ltþdt (from below and above,respectively), a and b approach 1þ 4K;tþdt=dK and trading volume declines. As a result, (25) and (26) approach zero both nearthe homogeneous case and at extreme heterogeneity.5

3.2. Comparative static analysis

The non-linear correspondence between RRAs and volume, in (25) and (26), can be plotted in three-dimensions, whereboth RRAs are measured along the X and Z axes, and volume on the vertical, Y axis. This makes a polynomial surface asheterogeneity increases along levels of dT and dC . An example for such surface is presented in Fig. 2, where volume is plottedgiven a 1% increase in the risky asset's price. In this benchmark case, WT ¼ 28,WC ¼ 100, r¼ 2%, and s ¼ 20%. The choice ofrelative wealth allocation, as well as the RRA combinations, is made to maximize the R2 of a linear regression with pro-portional shareholdings (time series average of pT ;t) as the dependent variable, across the 16 levels of heterogeneity; thismakes the model comparable with others in the literature, where the proportional weight of a given investor varies along alinear scale. Notice that once the RRAs, wealth, risk-free rate, and standard deviation of expected stock return parameters aredetermined, the market price of risk, as well as optimal asset allocations are implied by equilibrium conditions. To simplify,we assume spT ¼ 0 in this comparative static analysis. As in Dumas' (1989) model, our less risk-averse investor, type-T, holds anegative bond position while the type-C investor holds a positive bond position.

We expand RRAs in this section above and below l ¼ 2 since this value is supported empirically based on postWorldWar IIU.S. data. In particular, we let 0< dT <2:0 and 2< dC � 40. This choice of RRAs builds on common findings in the literature,ranging from 0.5 to about 4 or 5.6 We explore values as high as 40 (in this comparative static analysis only), as such high RRAsmay be justified based on recent survey findings by Kimball et al. (2008, KSS). They estimated that about 10% of agents haveRRAs higher than 100. Xiouros and Zapatero (2010) use a fitted G distribution based on KSS's findings, and allow a smallfraction of agents to have RRAs higher than 100. Xiouros and Zapatero use a harmonic mean risk aversion of 5.174 in thedifferent models they explore, and their highest RRA is 200, albeit shared by a very small fraction of agents.

Fig. 2 shows how volume varies along levels of dT , which declines along the Z-axis, and dC , which increases along the X-axis.As Fig. 2 shows, volume is miniscule when dT nears l from below, and when dC is marginally above l. Of particular interest isthe finding that volume increases with heterogeneity to a maximum and then declines toward zero when dT declines toward

5 Our simulations reveal that fT;tþdt and fC;tþdt are negative and small in absolute values at extreme homogeneity and heterogeneity. fT;tþdt is about �0.02 and fC;tþdt is about �0.05 to �0.15. Their absolute values increase at moderate heterogeneity, with averages of �0.21 and �0.52 for fT ;tþdt and fC;tþdt ,respectively.

6 Friend and Blume (1975), Mehra and Prescott (1985), Constantinides (2002), Cochrane (2009), and others conclude that RRA should range between 1and 3, although in more volatile periods it may exceed 4.


Fig. 2. Trading volume by investor type. Type-T trade Type-C trade.


zero and dC increases toward þ ∞. This property has material implications for turnover and liquidity, as we demonstrate inthe next section. Changes in share allocation between our two investors change the steepness and symmetry of the volumesurface, but not its fundamental form.

An intuitive explanation for zero trade in both highly homogeneous and highly heterogeneous markets builds on in-vestors' motivations for trade (ignoring 4K;tþdt as they approach zero in both extremes). In the homogeneous case, based on(18), the ratio ltþdt=dK is near unity for both agents; therefore, investors' trading strategies approximate buy-and-hold, thushardly trade. In the extreme heterogeneous case, investors’ trading preferences differ to an extent that makes bilateral tradeasymptotically zero. On the one hand, type-C investors are highly risk-averse with a small stock position (since l=dC/0), thusprice changes have negligible impact on their trades (i.e., a highly inelastic MTS). On the other hand, type-T investors have alarge stock position and a highly elastic MTS since l=dT is relatively large. Thus, a given change in price generates a largesupply or demand by type-T investors, but since type-C investors have little need in trading the stock, bilateral trading volumenears zero.

The previous analysis demonstrated that share allocation between both agents plays an important role in determining thelevel of bilateral volume, but share allocation was not the control variable in that analysis. Therefore, we explore next howvolume changes in five different combinations of RRAs (out of the 16 combinations mentioned above), ranging frommediumto extreme heterogeneity, while controlling for the change in share allocation. Using same benchmark parameters, we varythewealth level of type-T investors such that their proportional share holding changes from 2% to 100%, in fixed increments ofabout 5%. Total wealth of type-C investors remains 100 throughout, in all RRA combinations.

Fig. 3 shows on the X-axis the proportional shareholdings of type-T investors, and on the Y-axis, the turnover rate.Turnover is small when one type of the two investors dominates shareholdings. The solid line shows our benchmark scenario(Market 9), with RRAs dC ¼ 5:88 and dT ¼ 1:72. Maximum turnover of about 76% is achieved at about 37% shareholdings bytype-T investors. The more homogeneous scenarios lay below the solid line with lower volume that reaches a maximum at ahigher fraction of shareholdings by type-T investors. The dashed and dotted lines, above the benchmark solid line, representincreasingly heterogeneous scenarios, which reach a maximum turnover of about 120% when type-T investors hold a smallerfraction of the float versus the more homogeneous cases (about 15%).

The results suggest that heterogeneity and the distribution of share ownership between investor types have substantialeffects on turnover. Highest turnover levels can be achieved when preferences are more heterogeneous, and in general, asmaller fraction of type-T investors hold shares in the market, compared with homogeneous markets.

4. Calibration and simulation procedures

In this section, we explain the procedures and parameters we use to construct the simulated markets. The simulationsallow us to explore both time series and cross-sectional implications of heterogeneity. Unlike the previous analysis, here weconstruct the 16 RRA combinations (“markets”) by creating an RRA_factor, aiming to focus attention on less extreme RRA


Fig. 3. Bilateral trading volume vs. share allocation.


levels. The factor starts at 0.975 and declines 39 BPS from one combination to the next, until it equals 0.39. Because our RRAsmust bracket l, in each heterogeneity combination j we compute RRAs as follows:

dC;j ¼ 1:5� lREF.RRA Factorj;

and

dT ;j ¼ lREF � RRA Factorj;

where lREF is the reference against which RRAs lay above (type-C) or below (type-T). lREF is not used in any direct manner

when computing the l of simulated markets. We found that lREF ¼ 2.6 allows high homogeneity on one end of the scale, andreasonable heterogeneity on the other end of the scale. Consistent with the previous section, we construct the RRA combi-nations in this way since it yields a nearly perfectly linear change in proportional shareholdings between the two investortypes as heterogeneity increases from one market to the next. Further, since an investor's relative wealth also affects thelinearity of the cross-sectional regression, we compute the time series average of relative shareholdings and adjust WT ;t¼0

(given WC;t¼0 ¼ 100) to meet two goals: maximize the R2 of a regression of relative shareholdings across the 16 markets andmaximize the range of relative shareholdings across markets. Maximizing R2 is important to explore changes in relativeshareholdings along a linear scale; maximizing the range of relative shareholdings reveals more of the scale. With WT ;t¼0 ¼28, we obtain R2¼ 0.9956, and a scale that ranges from pC ¼ 23% to pC ¼ 67%. Increasing WT ;t¼0 above 28 increases R2

marginally but reduces the range, and vice versa. We verify that the boundaries on RRAs as detailed in Proposition 1 are notviolated.

The most homogeneous market emerges when the factor equals 0.975 (dC¼1.5*2.6/0.975 ¼ 4.00 and dT ¼ 2:6�0:975¼ 2.54) and the most heterogeneous market emerges when the factor equals 0.39 (dC¼1.5*2.6/0.39 ¼ 10.0 and dT ¼2:6� 0:39¼ 1.01), just above the log case. The 16 RRA combinations, as well as WA-RRA, are presented in Table 1. Notice thatexpected return increases monotonically with pC , while WA-RRA increases to a maximum, and then declines. It should benoted that spT varies across markets due to the different trading intensities. Therefore, in each simulated market, spT wascomputed recursively until it converged to the average value that emerges from the particular market. The average level ofmeasured spT s and their range in simulated markets are rather small, 0.78%e1.65% (see the right-most column of Table 1,Panel A).

In each of the heterogeneity combinations, we simulate 100 sample paths over 250 periods each. We assume that averageportfolio rebalancing is daily (dt¼ 1/250) in most tests, and in some tests also four trades per day (dt¼ 1/1000), striving toaccount for turbulent periods, or alternatively, for rapid professional and algorithmic traders. Such trading frequencies keepthe simulations consistent with the continuous-time framework of the model. The risky asset's price process is simulated by

dPt=Pt ¼ ðmt � 1=2s2Þdtþ sztffiffiffiffiffidt

p. The other parameters were chosen to approximate the U.S. post World War II averages, as

listed in Panel B of Table 1. Given wealth levels and the other parameters, Pareto-optimal share allocation across the twoagents is determined each period by iterative computation, until the mutually dependent Sharpe ratio and optimal share-holdings converge to equilibrium values.


Table 1Descriptive statistics and simulation parameters.

Panel A: Descriptive statistics

Market pC WA-RRA Expected Return Factor dT dC spT

Market 1 67% 3.553 15.6% 0.975 2.54 4.00 0.78%Market 2 65% 3.605 15.5% 0.936 2.43 4.17 0.95%Market 3 62% 3.657 15.4% 0.897 2.33 4.35 1.12%Market 4 60% 3.706 15.2% 0.858 2.23 4.55 1.27%Market 5 57% 3.748 14.9% 0.819 2.13 4.76 1.41%Market 6 55% 3.788 14.5% 0.78 2.03 5.00 1.53%Market 7 52% 3.819 14.0% 0.741 1.93 5.26 1.61%Market 8 49% 3.845 13.5% 0.702 1.83 5.56 1.65%Market 9 46% 3.845 12.8% 0.663 1.72 5.88 1.60%Market 10 43% 3.846 12.1% 0.624 1.62 6.25 1.63%Market 11 40% 3.831 11.3% 0.585 1.52 6.67 1.59%Market 12 37% 3.796 10.5% 0.546 1.42 7.14 1.54%Market 13 34% 3.741 9.7% 0.507 1.32 7.69 1.47%Market 14 30% 3.662 8.9% 0.468 1.22 8.33 1.40%Market 15 27% 3.555 8.1% 0.429 1.12 9.09 1.32%Market 16 23% 3.393 7.3% 0.390 1.01 10.00 1.21%

Panel B: Simulations parametersr¼ 2% r¼ 2%s ¼ 20% spT ¼ 0:78%� 1:65%N¼ 1.0 Q¼ 1.0

In Panel A, pC is the time-series average (250 periods) of the fraction of shares held by type-C investors. WA-RRA of dT and dC is weighted by pC and pT ¼ 1�pC ; Factor is used to expand the gap between RRAs; spT is the time series average of the standard deviation of pT ;t .In Panel B, the parameters used in the simulations roughly represent post-WW-II US financial market averages between January 1945 and December 2016.Equity premium 8%, riskless rate 2.0%, standard deviation of equities 20%; hence, l ¼ 2:0 and Sharpe ratio about 0.4. Stock and bond quantities arenormalized to 1.0. Sources: Robert Shiller's website: http://www.econ.yale.edu/~shiller/data.htm, Fama and French (1993), and Campbell et al. (2001).


5. Cross-sectional predictions for sharpe ratio variability

In this section, we return to the debate on the role that changes in heterogeneity play in causing variations in Sharpe ratio.We explore how the Sharpe ratio and its volatility vary along a cross-section of heterogeneity. In Xiouros and Zapatero (2010)and Chan and Kogan (2002), Sharpe ratio variability necessarily increases with heterogeneity, which is measured as thevariance across agents' RRAs. Because wemeasure trade as the smaller (in absolute terms) between both trade plans, trade isbounded not only in homogeneous markets, but in highly heterogeneous markets as well. This distinction from Chan andKogan and Xiouros and Zapatero's models yields a few predictions that we explore in the reminder of this section along thecross-section of average RRA.

5.1. Sharpe ratio vs. heterogeneity: level and range

Following the above, our model implies two main predictions on the association between heterogeneity and the level andrange of Sharpe ratio variability:

� Prediction 1: As heterogeneity increases, a marginal change in share allocation has a diminishing effect on marginalchange in the Sharpe ratio.

� Prediction 2: The time series range of average RRA would increase with heterogeneity.

Prediction 1 refers to the slope coefficient of a time series regression of average RRA on the Sharpe ratio, at given het-erogeneity levels. Prediction 2 builds on the finding from the comparative static analysis, where extremely heterogeneousstates allow more trade than extremely homogeneous states. Jointly, Predictions 1 and 2 imply that RRA variability wouldhave a small impact on Sharpe ratio variability in both extreme states, thus the highest impact in moderate heterogeneitystates. Table 2 and Fig. 4 show how the Sharpe ratio varies with WA-RRA. To minimize plausible non-stationarity biases andthe possibility that the less risk-averse investors dominate themarket (Wang,1996), we use five different sample paths of 250periods each (1250 “periods” in each market state). We conduct sensitivity analyses to two key parameters. First, in thebenchmark case, we assume a 20% standard deviation of the market index, representing normal times, and alternatively, weassume 35%, representing turbulent times. Second, we assume spT ¼ 0%, as well as spT ¼ 2% in all simulations to control for theimpact of the hedging demand. Thesemake four different panels aimed to explorewhether themodel predictions correspondwith the data. Table 2 shows summary statistics for all 16 markets, while, for clarity, the figure shows 8 of the 16 markets (1,and the even numbered markets 4 and above).

Panel A uses the benchmark parameters s¼ 20% and spT ¼ 0%. Panels B to D show how the association between Sharperatio and WA-RRA varies with our control parameters. Of primary interest are the range 2e5 for WA-RRA and the range


http://www.econ.yale.edu/%7Eshiller/data.htm

Table 2Sharpe ratio vs. heterogeneity: return volatility 20% and 35%, with and without hedging demand.

Panel A: s ¼ 20%, spT ¼ 0%

Panel A Sharpe ratio WA-RRA Regression

1 2 Average 3 STD 4 High 5 Low 6 Average 7 STD 8 High 9 Low 10 Slope

Market 1 0.685 0.005 0.694 0.672 3.598 0.021 3.640 3.542 0.224Market 2 0.678 0.007 0.692 0.661 3.640 0.030 3.700 3.560 0.220Market 3 0.669 0.009 0.687 0.647 3.680 0.041 3.762 3.574 0.214Market 4 0.656 0.011 0.679 0.628 3.717 0.053 3.825 3.580 0.206Market 5 0.638 0.013 0.666 0.606 3.745 0.067 3.881 3.574 0.195Market 6 0.617 0.015 0.649 0.580 3.770 0.082 3.940 3.561 0.181Market 7 0.591 0.017 0.628 0.551 3.786 0.099 3.993 3.537 0.167Market 8 0.562 0.018 0.601 0.520 3.798 0.117 4.046 3.506 0.150Market 9 0.526 0.018 0.567 0.484 3.783 0.135 4.075 3.451 0.133Market 10 0.491 0.018 0.532 0.450 3.770 0.154 4.106 3.398 0.115Market 11 0.453 0.017 0.494 0.415 3.744 0.171 4.123 3.334 0.099Market 12 0.415 0.016 0.453 0.381 3.699 0.187 4.120 3.257 0.083Market 13 0.376 0.014 0.410 0.346 3.638 0.201 4.096 3.168 0.068Market 14 0.338 0.012 0.367 0.312 3.555 0.212 4.044 3.064 0.055Market 15 0.300 0.010 0.324 0.279 3.447 0.221 3.961 2.943 0.043Market 16 0.261 0.007 0.280 0.245 3.287 0.224 3.813 2.781 0.033

Panel B: s ¼ 20%, spT ¼ 2%

Panel B Sharpe ratio WA-RRA Regression % Change vs. Panel A

1 2 Average 3 STD 4 High 5 Low 6 Average 7 STD 8 High 9 Low 10 Slope 11 Sharpe ratio 12 WA-RRA 13 Slope

Market 1 0.687 0.006 0.699 0.672 3.608 0.026 3.661 3.539 0.225 0.3% 0.3% 0.5%Market 2 0.680 0.008 0.698 0.659 3.650 0.038 3.725 3.552 0.221 0.3% 0.3% 0.5%Market 3 0.671 0.011 0.693 0.643 3.687 0.050 3.788 3.557 0.215 0.2% 0.2% �0.2%Market 4 0.656 0.013 0.685 0.623 3.718 0.065 3.850 3.550 0.205 0.0% 0.0% �0.2%Market 5 0.636 0.016 0.670 0.598 3.734 0.082 3.902 3.526 0.193 �0.3% �0.3% �0.9%Market 6 0.612 0.018 0.651 0.569 3.741 0.100 3.948 3.490 0.178 �0.8% �0.8% �1.9%Market 7 0.582 0.019 0.625 0.537 3.732 0.118 3.980 3.438 0.161 �1.5% �1.4% �3.2%Market 8 0.549 0.019 0.594 0.504 3.712 0.136 4.001 3.376 0.143 �2.2% �2.3% �4.6%Market 9 0.510 0.019 0.555 0.466 3.662 0.153 3.992 3.288 0.125 �3.0% �3.2% �5.9%Market 10 0.473 0.018 0.516 0.432 3.613 0.169 3.981 3.204 0.107 �3.6% �4.2% �7.0%Market 11 0.435 0.017 0.475 0.397 3.555 0.184 3.960 3.112 0.091 �4.0% �5.0% �7.7%Market 12 0.398 0.015 0.435 0.364 3.487 0.199 3.931 3.013 0.076 �4.1% �5.7% �7.9%Market 13 0.361 0.014 0.395 0.331 3.413 0.214 3.897 2.906 0.063 �3.9% �6.2% �7.6%Market 14 0.326 0.012 0.355 0.300 3.329 0.231 3.858 2.789 0.051 �3.6% �6.3% �6.8%Market 15 0.291 0.010 0.317 0.269 3.232 0.247 3.808 2.661 0.041 �3.0% �6.2% �5.7%Market 16 0.255 0.008 0.276 0.237 3.095 0.263 3.719 2.498 0.032 �2.4% �5.8% �4.3%

Panel C: s ¼ 35%, spT ¼ 0%

Panel C Sharpe ratio WA-RRA Regression

1 2 Average 3 STD 4 High 5 Low 6 Average 7 STD 8 High 9 Low 10 Slope

Market 1 1.204 0.017 1.241 1.162 3.611 0.043 3.703 3.503 0.395Market 2 1.195 0.024 1.248 1.137 3.659 0.061 3.790 3.505 0.390Market 3 1.182 0.031 1.253 1.107 3.706 0.082 3.883 3.502 0.381Market 4 1.162 0.039 1.254 1.070 3.753 0.107 3.985 3.490 0.368Market 5 1.135 0.047 1.249 1.028 3.792 0.135 4.089 3.464 0.351Market 6 1.101 0.055 1.238 0.980 3.832 0.167 4.204 3.432 0.331Market 7 1.061 0.062 1.220 0.928 3.865 0.201 4.322 3.389 0.307Market 8 1.013 0.068 1.193 0.874 3.897 0.240 4.452 3.341 0.281Market 9 0.954 0.071 1.150 0.813 3.907 0.281 4.571 3.271 0.252Market 10 0.895 0.072 1.101 0.756 3.922 0.322 4.700 3.206 0.223Market 11 0.831 0.071 1.039 0.698 3.926 0.365 4.824 3.133 0.194Market 12 0.764 0.068 0.965 0.642 3.914 0.407 4.931 3.051 0.165Market 13 0.694 0.062 0.880 0.585 3.887 0.447 5.022 2.960 0.137Market 14 0.624 0.054 0.788 0.530 3.837 0.484 5.081 2.857 0.111Market 15 0.554 0.046 0.691 0.476 3.762 0.516 5.098 2.741 0.088Market 16 0.480 0.036 0.587 0.419 3.629 0.538 5.030 2.590 0.067

Panel D: s ¼ 35%, spT ¼ 2%

Panel D Sharpe ratio WA-RRA Regression % Change vs. Panel C


Market 1 1.193 0.018 1.221 1.145 3.583 0.047 3.656 3.456 0.384 �0.9% �0.8% �2.7%Market 2 1.178 0.025 1.218 1.113 3.615 0.067 3.719 3.437 0.374 �1.4% �1.2% �4.0%Market 3 1.158 0.032 1.211 1.076 3.643 0.089 3.782 3.408 0.360 �2.0% �1.7% �5.5%

(continued on next page)



Table 2 (continued )

Panel D: s ¼ 35%, spT ¼ 2%

Panel D Sharpe ratio WA-RRA Regression % Change vs. Panel C


Market 4 1.130 0.039 1.196 1.033 3.663 0.115 3.844 3.365 0.340 �2.8% �2.4% �7.4%Market 5 1.093 0.046 1.171 0.984 3.670 0.142 3.897 3.306 0.318 �3.7% �3.2% �9.5%Market 6 1.049 0.050 1.138 0.931 3.667 0.172 3.945 3.236 0.291 �4.8% �4.3% �11.9%Market 7 0.997 0.053 1.094 0.876 3.649 0.201 3.979 3.155 0.263 �6.0% �5.6% �14.4%Market 8 0.940 0.054 1.039 0.819 3.619 0.230 4.001 3.064 0.233 �7.3% �7.1% �17.1%Market 9 0.872 0.052 0.970 0.758 3.558 0.256 3.988 2.952 0.202 �8.6% �8.9% �19.9%Market 10 0.807 0.049 0.899 0.703 3.494 0.278 3.963 2.846 0.174 �9.8% �10.9% �22.3%Market 11 0.742 0.044 0.823 0.648 3.416 0.296 3.916 2.735 0.147 �10.8% �13.0% �24.3%Market 12 0.677 0.038 0.748 0.596 3.325 0.311 3.848 2.619 0.123 �11.4% �15.1% �25.7%Market 13 0.613 0.033 0.674 0.544 3.225 0.324 3.772 2.499 0.101 �11.7% �17.0% �26.3%Market 14 0.552 0.028 0.606 0.495 3.117 0.336 3.705 2.374 0.082 �11.4% �18.8% �26.1%Market 15 0.494 0.023 0.540 0.446 3.001 0.349 3.633 2.241 0.066 �10.8% �20.2% �24.9%Market 16 0.433 0.019 0.471 0.395 2.849 0.361 3.533 2.076 0.051 �9.8% �21.5% �23.0%


0.3e0.5 for Sharpe ratio. Extreme values may correspond to Ludvigson and Ng (2007), who implemented dynamic factoranalysis and estimated quarterly conditional Sharpe ratios from as low as �0.1 to þ1.6, between 1960 and 2003.

Fig. 4. Sharpe ratio vs. weighted-average RRA.


Fig. 4. (continued).


The first finding worthy highlighting is that in all panels, the Sharpe ratio hardly varies with WA-RRA in the most ho-mogeneous case (Market 1). This finding confirms Prediction 1, and is the result of miniscule trade (i.e., little share redis-tribution). However, the same result obtains in the most heterogeneous case (Market 16). Panel A of Table 2 shows that thestandard deviation of Sharpe ratio (column (3)) is almost zero in the few extreme markets, both homogeneous and het-erogeneous. However, the extremely homogeneous and heterogeneous markets differ with respect to the dispersion of WA-RRA (column (7)): in the homogeneousMarket 1,WA-RRA hardly varies (Std. Dev.¼ 0.021), while inMarket 16 its variability isabout an order of magnitude higher (Std. Dev.¼ 0.224), althoughWA-RRA is lower in Market 16 (column (6)): 3.287 vs. 3.598in Market 1. This finding supports Prediction 2. Importantly, the large variation of WA-RRA in Market 16 hardly affects theSharpe ratio level (column (10)): the regression slope between the Sharpe ratio (dependent) and WA-RRA, is 0.033, thesmallest across all markets.

Sharpe ratio does not vary with WA-RRA in Market 16 because type-T investors, who hold on average about 77% of shareswith dT ¼ 1:01, experience little interest in trade by their counter party, type-C investors. Type-C's high RRA implies that theirallocation to shares is small, and coupled with their small share in aggregate shareholdings, their rebalancing needs are smallrelative to the quantities that type-T would have wanted to trade.

The model implies that in extreme homogeneity or heterogeneity, changes in share distribution would not change themarket price of risk materially either because the changes in WA-RRA are too small (Market 1) or because of the absence of acounter party for trade, which make the regression slope nearly zero (Market 16). These findings are consistent with thecombined effects of Predictions 1 and 2.

While the lowest and highest levels of WA-RRA (columns (8) and (9)) in Market 16 may be consistent with those assumedin the literature, 2.781 and 3.813, the corresponding level of Sharpe ratio in this market is rather low, 0.261. In contrast,average Sharpe ratio (column (2)) inMarket 1 is high, 0.685, relative to accepted estimates of 0.3e0.5. In Panel A of Table 2, the



heterogeneity structures that roughly seem to correspond to the Sharpe ratio range 0.3e0.5 appear to be Markets 10 to 15. Inthese markets, WA-RRA obtains values of 3.447e3.770, and the corresponding Sharpe ratios range from 0.300 to 0.491. Thesummary statistics in Panel A of Table 2 show that the regression slope (column (10)) of Market 10, for example, is 0.115, andthe difference between the highest and lowestWA-RRA is 0.708 (¼4.106e3.398). This gap is associatedwith a 0.082 gap in theSharpe ratio of that Market 10 (¼0.532e0.450). This makes an 18.2% increase in Sharpe ratio from the lowest to highest statesof Market 10. Such variation in Sharpe ratio, caused by a rather small variation inWA-RRA, may be instrumental in associatingstock price variability with share redistribution through heterogeneity.

Panel B of Table 2 and Panel B of Fig. 4 present the association betweenWA-RRA and the Sharpe ratio, along with hedgingdemand, a given spT ¼ 2% in all simulated markets. Panel B of the table shows on columns (11)e(13) the percentage changeversus Panel A of Sharpe ratio, WA-RRA, and the regression slope, respectively. Column (13) shows that the regression slopecoefficient of the most homogeneous, Market 1, is 0.5% higher, while the most heterogeneous, Market 16, has a�4.3% smallerslope. Market 12, with RRAs dT ¼ 1:42 and dC ¼ 7:14 (average 3.80) exhibits the greatest decline in regression slope, �7.9%versus the corresponding case. This RRA combination also exhibits the greatest decline in Sharpe ratio (column (11)), from0.415 to 0.398, or �4.1%. Nevertheless, the greatest decline in WA-RRA (column (12)) occurred in Market 14 (�6.3%). Thesefindings highlight the importance of hedging against changes in WA-RRA in market states with high trading activity.

Next, we explore the role that return volatility plays at different levels of heterogeneity and trade. The exogenously givenexpected standard deviation affects the magnitude of portfolio rebalancing. While fixed in the model, it may vary both overtime and in the cross-section of stocks for different reasons. For example, Barinov (2014) shows that market-wide volatilityincreases with turnover, after controlling for a variety of documented factors, and Weinbaum (2009) shows that heteroge-neity and rebalancing frequency increase stock return variability.

Regression results in Panel C of Table 2 are based on same parameters used for Panel A except for replacing the 20%standard deviation of the market index with 35%. In this scenario, Markets 14e16 obtain the smallest Sharpe ratio values,which range from 0.480 to 0.624 (column (2)). In the more homogeneous markets, Sharpe ratio exceeds 1.0. Recall that thesehigh values represent highly volatile periods, not the long-run average. The variation inWA-RRA in thesemarkets ranges from3.629 to 3.837 (column (6)) with more than doubled standard deviations (column (7)), and about double regression slopes(column (10)) comparedwith those shown in Panels A and B. These findings suggest that in turbulent times not onlymayWA-RRA variability double, but a given change has about twice as much impact on Sharpe ratio. In Panel D we repeat the analysisconducted in Panel C but introduce hedging demand with spT ¼ 2%. The results show that Sharpe ratio levels (column (2)) inMarkets 14e16 range from 0.433 to 0.552, about 10% lower than the comparable levels in Panel C, with about 25% smallerregression slopes (column (13)), albeit those slopes are still about 50% higher than those in Panels A and B. These findingsindicate that while the importance of hedging against RRA variability increases in highly volatile and actively traded states,Sharpe ratio variability is still higher than the benchmarks.

Overall, it appears that the model may generate sizable variation in Sharpe ratio as WA-RRA varies due to redistribution ofthe risky asset's holdings across our two investor types. At extreme homogeneity and heterogeneity, we find little or noSharpe ratio variation. Nevertheless, Sharpe ratio is sensitive to changes in WA-RRA at moderate heterogeneity levels, con-firming Prediction 1. Prediction 2 is partially confirmed: while the regression slope declines monotonically with heteroge-neity, its effective impact is maximized at moderate heterogeneity. Lastly, we note that the hedging component plays a ratherimportant role when s ¼ 20%, representing the historical U.S. average, and its impact further increases with volatility andtrade intensity.

5.2. The cross-section of sharpe ratio vs. its risk

�Prediction 3: Sharpe ratio variability would increase with heterogeneity to a maximum, and then decline.Panel A of Fig. 5 shows how the Sharpe ratio and its standard deviation vary across levels of heterogeneity. The locus

features Sharpe ratio levels between 0.261 in themost heterogeneousMarket 16 and 0.685 in the homogeneousMarket 1. Thehighest variability in Sharpe ratio occurs in Market 9, which is one of the most liquid markets. The plot highlights that themore homogeneous markets, Markets 1e9, yield a higher Sharpe ratio for a given level of Sharpe ratio variability, thusrepresent the better choice from a central planner's perspective.

However, this result may point at what appears to be a “cost of liquidity”: a more liquid market facilitated more shareredistribution across agents. Therefore,WA-RRAvariability increases and induces higher Sharpe ratio volatility, which implieshigh stock return variability. In this respect, high stock return variability can be considered a cost of liquidity at the aggregatelevel. This aggregate cost of liquidity notion is presented in Panel B of Fig. 5. Unlike the previous conclusionwhereby themorehomogenous markets are preferable, here the more heterogeneous markets generate higher turnover (and liquidity), for agiven level of Sharpe ratio variability in Markets 9e16. These findings correspond with Prediction 3, and point at theaggregate cost of liquidity.7

7 We thank Yakov Amihud and Fernando Zapatero for discussions on that implication of the model.


Fig. 5. Sharpe ratio, its risk, and annual turnover rates.


6. Cross-sectional predictions for turnover and liquidity

In this section, we explore how levels and volatilities of turnover and liquidity vary across levels of heterogeneity. All time-series simulations are conducted in a way similar to the previous analysis, except that in these analyses the averages andmedians are measured across 100 sample paths of 250 periods in each market state. The results incorporate the market-specific spT that was computed recursively until convergence. Prediction 4 explores the linkages between heterogeneityand market activity:

�Prediction 4: Turnover rates, liquidity, and their risks would increase with heterogeneity to a maximum, and then decline.

6.1. The cross-section of turnover and its risk

Over the past several years, annual turnover rates increased in many stock exchanges worldwide. Table 3 shows theaverage turnover in a number of exchanges in 2015 and 2016, as reported by theWorld Federation of Exchanges (2015) (WFE).The two Chinese exchanges, in Shenzhen and Shanghai, facilitated turnover of 519% and 450% in 2015 but 374% and 192% in2016, respectively. A few other exchanges reported turnover rates between 100% and 200%. By NYSE statistics,8 averagemonthly turnover during 2015 varied between 56% and 64%, and in 2016 between 53% and 84%. Barber and Odean (2000) findthat average individual investor's annual turnover rate exceeds 100%. In a theoretical model with heterogeneous preferences,Weinbaum (2009) finds miniscule trading volume: to obtain volume that is close to real markets, his representative investors

8 Source: NYSE Facts and Figures. Retrieved 5/28/2017. Available at: http://www.nyxdata.com/nysedata/asp/factbook/viewer_edition.asp?mode¼table&key¼3149&category¼3.


http://www.nyxdata.com/nysedata/asp/factbook/viewer_edition.asp?mode=table&key=3149&category=3







Table 3Average turnover rates across selected exchanges: 2015 and 2016.

Exchange 2016 Average 2015 Average

AmericasBM&FBOVESPA S.A. 79.5% 85.6%Bolsa de Comercio de Santiago 11.2% 10.3%Bolsa de Valores de Colombia 13.5% 13.5%Bolsa Mexicana de Valores 28.2% 25.8%TMX Group 63.1% 68.9%Asia - PacificAustralian Securities Exchange 64.1% 63.2%Bursa Malaysia 26.9% 29.1%Hochiminh Stock Exchange 39.1% 36.0%Hong Kong Exchanges and Clearing 42.2% 65.0%Indonesia Stock Exchange 22.4% 21.2%Japan Exchange Group 116.8% 113.8%Korea Exchange 129.0% 149.8%National Stock Exchange of India Limited 46.1% 44.1%NZX Limited 13.2% 12.2%The Philippine Stock Exchange 14.5% 16.1%Shanghai Stock Exchange 192.4% 449.7%Shenzhen Stock Exchange 374.2% 518.6%Singapore Exchange 31.9% 30.9%The Stock Exchange of Thailand 80.9% 77.8%Taipei Exchange 171.0% 196.0%Taiwan Stock Exchange 59.0% 75.7%Europe - Middle East - AfricaAthens Stock Exchange 38.5% 42.6%BME Spanish Exchanges 97.8% 124.3%Borsa Istanbul 168.6% 185.2%Deutsche B€orse AG 74.9% 84.2%Dubai Financial Market 40.7% 49.1%The Egyptian Exchange 39.2% 26.7%Euronext 52.5% 62.0%Irish Stock Exchange 21.1% 16.4%Johannesburg Stock Exchange 38.4% 41.3%Moscow Exchange 25.7% 29.8%Nasdaq Nordic Exchanges 54.9% 56.4%Oslo Børs 45.2% 49.5%Saudi Stock Exchange (Tadawul) 77.5% 103.8%SIX Swiss Exchange 60.7% 62.9%Tel-Aviv Stock Exchange 23.4% 23.1%

Source: World Federation of Exchanges, Retrieved: December 6th, 2015


must trade about 100 times each day. The discrepancy between empirical findings and theoretical models of turnover attractsmuch research attention, partly because turnover is related to liquidity, and liquidity is a priced factor [e.g., Amihud et al.(2015), and the references therein].

We next examine whether our heterogeneity structure can help explain such high levels of turnover, and how turnoverand liquidity are associated with Sharpe ratio variability. We measure annual turnover rate by normalizing volume by thenumber of total shares outstanding at the end of the period, and dividing by dt,

Annual Turnover% ¼ 1250� dt

X250t¼1

Min��VOLðCÞt��; ��VOLðTÞt��Nt : (27)

An important parameter affecting the turnover rate is the average trading frequency. Given the continuous-time frame-work of the model, we restrict attention to daily and four times daily frequencies. Additional parameters that affect turnover,across all heterogeneity levels, are the standard deviation of stock returns, s, and spT .

The prediction concerning turnover builds on our earlier results and the comparative static analysis. Indeed, as the threepanels of Fig. 6 show, the annual turnover rate reaches amaximum atmoderate heterogeneity levels and then declines.9 PanelA shows the benchmark case, with dt¼ 1/250 and s ¼ 20%, while spT obtains the market-specific level that was computedrecursively, as in Panel A of Table 1. Turnover increases to a maximum of 56%, rather close to the NYSE averages of 2015 and2016, at heterogeneity level 1.62e6.25 (Market 10), with WA-RRA 3.846. Recall that the above simulation measures

9 This decline would have reached zero, rather symmetric to the homogeneous case, if we had allowed 0< dT <1 and increased dc to 40.


Fig. 6. Annual turnover rate across levels of heterogeneity. Annual turnover rate across 16 levels of heterogeneity. WT ;t¼0 ¼ 28, WC;t¼0 ¼ 100, r¼ 2%, spT variesby market.


incorporate the assumption that investors hedge against changes in WA-RRA, although it is unclear whether this assumptionholds in practice.

Panel B of Fig. 6 shows that if rebalancing frequency is four times per day, i.e., dt¼ 1/1,000, and s ¼ 35% with no hedgingdemand, turnover rate exceeds 180%, at the heterogeneity level 1.52e6.67 (Market 11) with WA-RRA 3.831. To the best of ourknowledge, no existing asset pricing model that incorporates turnover based on heterogeneity of preferences reported suchhigh turnover rate levels. Overall, Prediction 4 holds in the cross-section of turnover.

6.2. The cross-section of illiquidity and its risk

Next, we examine cross-sectional patterns of illiquidity and its risk by heterogeneity. If illiquidity varies systematically inthe cross-section of heterogeneity, then heterogeneity may be a latent factor that affects the illiquidity premium. For het-erogeneity to affect cross-sectional measures of illiquidity, average RRA should vary in the cross-section as well. Amihud(2002) suggested the following time series measure for illiquidity:



ILLIQ ¼ 1D

XDt¼1

jrt j$Volt

; (28)

where jrt j is the absolute rate of return at t, $Volt is the monetary value of trade during t, and D is the number of trading
periods in the sample. In terms of our model, $Volt is the product of price and bilateral traded quantity: $Volt ¼ Pt �Minð��VOLðCÞt��; ��VOLðTÞt��Þ. For a given price and a given absolute return, volume may be higher in one stock than in another,either due to a high turnover rate or firm size (see Brennan et al., 2013). To control for size, which is not a relevant attribute inour framework, we scale Amihud's original ILLIQ measure by the number of shares outstanding. This yields a modifiedmeasure, scaled by 102, denoted ILLIQ_M,
ILLIQ M ¼ 100250� dt

X250t¼1

jrt jPt �Min

��VOLðCÞt��; ��VOLðTÞt��Nt: (29)

The reciprocal of (29) can be interpreted as the price elasticity of trade, thus expected to be sensitive to the more inelasticMTS between type-C and type-T investors. The reason is that we consider the minimum between

��VOLðCÞt�� and ��VOLðTÞt�� inthe denominator of (29) as the relevant amount of bilateral trade. This notion is comparable to theoretical and empiricalmeasures of liquidity, such as Johnson (2006), where the absolute proportional quantity traded is divided by the absolute rateof return of the stock.

Panel A of Fig. 7 shows how ILLIQ_M varies along the cross-section of heterogeneity at daily and 4 times daily rebalancingfrequencies. Illiquidity is high in the extremely homogeneous and extremely heterogeneous markets, in an asymmetric way,due to the asymmetric pattern of turnover, as discussed in the comparative static analysis. The Figure highlights the interiorminimum for illiquidity in both rebalancing frequencies: it occurs at Market 7 if rebalancing is daily (WA-RRA¼ 3.819), and atMarket 12 for 4 times daily rebalancing (WA-RRA¼ 3.796).

Recall that upon price declines, our type-T investors sell shares to type-C investors, and conversely upon price increases,thus heterogeneity is perfectly correlated with expected stock returns. For example, if heterogeneity declines from theminimum illiquidity in Market 7 (daily rebalancing), to the more homogeneous Market 5, WA-RRA declines from 3.82 to 3.75,about�1.8%, and pC increases from 57% to 62%. This shift increases illiquidity from 5.24 to 5.62, or 7.3%, fourfold the change inWA-RRA. Symmetrically, about similar decline in WA-RRA, but through a shift toward greater dominance of type-T investors(following stock price increases), makes a shift fromMarket 7 toMarket 13. In this caseWA-RRA also declines about 1.9%e3.74and pC declines from 57% to 38%. Yet, illiquidity increases about 57%, from 5.24 to 8.22, highlighting the sensitivity of the levelof illiquidity to changes in heterogeneity.

6.3. Liquidity risk

Several researchers explore the relation between liquidity risk and different measures of volume. Acharya and Pedersen(2005) and Johnson (2008) incorporate liquidity risk in asset pricing models. Nevertheless, the association between the levelof liquidity and liquidity risk has not been explored in the context of heterogeneous preferences. Essentially, it is unclearwhether under heterogeneous preferences a reduction in the level of illiquidity is necessarily associated with lower illiquidityrisk. Panel B of Fig. 7 shows how median illiquidity, measured on the y-axis, is associated with the standard deviation ofilliquidity, on the x-axis. At the bottom of the V-shaped pattern is Market 7, with standard deviation of ILLIQ_M¼ 7.0 andILLIQ_M¼ 5.24. The RRA parameters of Market 7 appear reasonable from the asset pricing literature perspective, as detailed inTable 1. Higher heterogeneity than that of Market 7 is associated with higher illiquidity risk. Yet, higher homogeneity isassociated with about equal illiquidity risk, but higher illiquidity.

We note that Prediction 4 holds for the association between liquidity and its risk, presented in Panel A of Fig. 7, as well asthe association between turnover and its risk. Together with the previous predictions and findings that highlight themagnitude of changes in average RRA due to heterogeneity on Sharpe ratio, its volatility, and on liquidity, we conclude thatthe role of heterogeneity should not be underestimated.

7. Conclusions

In this paper, we analyze the extent to which heterogeneity motivates trade in an asset pricing model under informationsymmetry with time-separable, power utilities. Two uniquely defined investors have RRAs that bracket the market price ofrisk, where heterogeneity is measured by the dispersion between both RRAs. Investors’ optimal portfolio rules implyintertemporal bilateral trade, thus average RRA in themarket changes stochastically, but is perfectly correlated with expectedstock returns. Using the martingale approach, we solve for the hedging component due to this additional state variable.

Our key findings stem from a simple observation: while turnover and liquidity increase with heterogeneity, at some level,further widening the gap between both RRAs reduces volume and market activity. The reason is that the less risk-averseinvestor holds mainly equity, while the more risk-averse investor would hold mostly bonds. Only at moderate heterogene-ity levels would wealth redistribution cause large and countercyclical variations in Sharpe ratio that may justify the


Fig. 7. Cross-sectional implications of heterogeneity.


empirically observed high stock price volatility. In such states, liquidity reaches its highest levels, but since high liquidity iscoupled with a high Sharpe ratio and hence stock return volatility, it is unclear whether maximizing liquidity is indeeddesired. In that sense, the high volatility of the equity premium affects stocks' betas, therefore can be regarded as the cost ofhigh liquidity.

Appendix 1

A brief literature review

Motivated by the gap between theory and practice, Wang (1994), Acharya and Pedersen (2005), Lo and Wang (2006),Johnson (2006, 2008), and others, developed asset pricing models with volume, liquidity, and liquidity risk. Volume in thesecontributions stems from exogenous perturbations and in some cases from asymmetric information, yet turnover rates,where analyzed, are rather small. An alternative path to generate volume is heterogeneous preferences, pioneered by Dumas(1989), featuring exogenous stock prices, as didWang (1996). Both papers focus on the term structure and use time-separableutility functions for two agents. Like Dumas, we take the stock prices process as given, and solve for bilateral volume; unlikeDumas, we focus on stock market returns, liquidity, and trading activity, taking bond return as given.

Chan and Kogan (2002) argue that given heterogeneous preferences, variation in agents' relative wealth would result insufficient variation in average relative risk aversion (RRA) to explain empirical anomalies. In particular, it may serve to justifythe variation in RRA needed in Campbell and Cochrane's (1999) model to explain Sharpe ratio volatility. Yet, Xiouros and



Zapatero (2010) calibrate heterogeneity empirically and find that the Sharpe ratio variability in Campbell and Cochrane's(1999) model is too high to be due to heterogeneity.

Empirically, Fama and French (1988), Campbell and Shiller (1988a,b), and Campbell (1991) point at changes in the equitypremium as one of the key contributors of price/dividend ratio volatility. The volatility decomposition approach (Cochrane,1992; Campbell and Mei, 1993; Chen and Zhao, 2009) distinguishes between cash flow news and discount rate news as ifexpressing the asset's price in a discounted cash flow setup. The findings generally give substantially moreweight to discountrate risk. Consistently, our model solely accounts for discount rate risk, as it varies due to changes in average RRA.

Appendix 2

Derivation of optimal asset allocation

Derivation of the optimal asset allocation follows from equations (7) and (8) (e.g., Cvitanic and Zapatero, 2004; Pennacchi,2008). Investors' current wealth is the present value of expected dividends through lifetime, plus the present value of ter-minal wealth. Because the dividend stream equals consumption, we have:

Wt ¼ Et

264ZTt

Ms

Mtcsdsþ

MTMt

WT

375; (A1)

where T is the terminal date. (A1) may be considered an intertemporal budget constraint. Converting (A1) to a static structure

with the Lagrange multiplier bl, we have:

Maxcs cs2½t;T �;WT

¼ Et

264Zt

TUðcs; sÞdsþ HðWT ; T�375þ bl

264MtWt � Et

0B@Zt

TMscsdsþMTWT

1CA375: (A2)

The first-order condition for consumption at all s is:

vUðcs; sÞvcs

¼ blMs; (A3)

and at T is:

vH�WT ; T

�vWT

¼ blMT : (A4)

�1 �1
Define two inverse functions, for each of the FOCs above, as GU ¼ ½vU=vc� and GH ¼ ½vH=vW � . Then,
c*s ¼ GU

�blMs; s�; (A5)

W* ¼ G�blM ; T

�: (A6)

T H T

Substituting (A5) and (A6) into (A1) we have:

Wt ¼ Et

264ZTt

Ms

MtGU

�blMs; s�dsþMT

MtGH

�blMT ; T�375: (A7)

Because the wealth process can be interpreted as a representation of the dividend process, which equals consumption, itsatisfies a Black-Scholes-Merton PDE, given the distribution ofMs and the structure of the utility function. Because dzt and dztare perfectly correlated, equation (4) in the text can be written as:

dpT ¼ mpðpT ; tÞdt þ spdzt : (A8)

The martingale properties of (10) in the text, together with (A7) and (A8), make the optimal allocation to the risky asset afunction ofMt , t, pT ;t , and T . The reasonWt depends not only on, t, and T but on pT ;t as well, is that the expectation in equation



(A7) depends on the distribution of future values of the pricing kernel. Equations (7) and (8) in the text show that thisdistribution depends on the initial value, Mt , as well as on Qt , which may vary with pT ;t .

By Ito lemma, WðMt ;pT ;t ; TÞ follows the process:

dW ¼ WMdM þWpT dpT þ vWvt

dt þ 12WMMðdMÞ2 þWMpT

ðdMÞðdpT Þ þ12WpTpT ðdpT Þ2; (A9)

which can be reorganized as:

dW ¼�� rMWM þ mpWpT þ

vWvt

þ 12Q2M2WMM �QspMWMpT

þ 12s2pWpTpT

�dt

þ½spWpT �QMWM �dz≡ AWdt þ BWdz:

(A10)

Expected wealth growth equals the riskless rate in addition to the risk premium,

AW þ GU

�blMt ; t�¼ rWt þ BWQ; (A11)

from which we have the PDE:

0 ¼ GU

�blMt ; t�þ�Q2 � r

�MWM þ ðmp � spQÞWpT �QspMWMpT

þvWvt

þ 12Q2M2WMM þ 1

2s2pWpTpT � rW ;

(A12)

subject to WðMT ;pT ;T ;TÞ ¼ GHðblMT ;TÞ. The solution to this problem, WðMt ;pT ;t ;t; blÞ ¼ Wt , determines bl as a function of Mt ,

Wt , and pT ;t . Based on market completeness, we can replicate the wealth process and the dividends/consumption process.Thus, equating the coefficients of (A10) with those of (5) in the text reveals that BW ¼ atWs ¼ spWpT �QMWM . Solving for at,we obtain:

at ¼ �QMWM

WsþWpTsp

Ws: (A13)

Recalling from (11) that ðm� rÞ=s ¼ Q, and replacing for Q in (A13), we have:

at ¼ �MWM

Wm� rs2

þWpTspWs

: (A14)

This structure is analogous to the structure one obtains from the dynamic stochastic programming approach where, underperfect correlation between dzt and dzt , MWM ¼ JW=JWW and WpT ¼ � JWpT

=JWW .

Appendix 3

Proof of Proposition 1. Optimum intertemporal trade

Because ltþdtþ4K;tþdt

ltþ4K;ty1, the term in the square brackets in the numerator of (25) is approximately Ly1� ðltþdt þ

4K;tþdtÞ=dK . L is positive if dC > ltþdt þ 4C;tþdt , in which case (24) has a negative slope, implying that an increase in the stockprice is associated with selling some shares (type-C, contrarian). Symmetrically, L is negative if dT < ltþdt þ 4T ;tþdt; thus, givena price increase, type-T investors will buy some shares, acting like positive feedback, or trend-chasing traders.

Appendix 4

Proof of Proposition 2. The drivers of volumeUsing (26) and replacing NT ;tþdt ¼ NT ;t þ dNT ;tþdt and NC;tþdt ¼ Ntþdt � NT ;t � dNT ;tþdt , one obtains:



NtþdtPtþdt ¼ltþdt þ 4T;tþdtT

dT

��NT;t þ dNT ;tþdt

�Ptþdt þ DT;tþdt

�þltþdt þ 4C;tþdt

dC

��Ntþdt � NT;t � dNT;tþdt

�Ptþdt þ DC;tþdt

�:

(B1)

Solving for optimal trade by type-T, dNT ;tþdt ,

dNT;tþdt ¼NtþdtPtþdtð1� bÞ � NT;tPtþdtða� bÞ � aWT;tþdtð1� aÞ � bWC;tþdtð1� bÞ

Ptþdt

hltþdt

�1dT� 1

dC

�� 4C;tþdt

dC þ 4T;tþdt

dT

i ; (B2)

where a≡ðltþdt þ 4T ;tþdtÞ=dT , and b≡ðltþdt þ 4C;tþdtÞ=dC .Using same procedure, replace NC;tþdt ¼ NC;t þ dNC;tþdt and NT ;tþdt ¼ Ntþdt � NC;t � dNC;tþdt in (26) and solve for the

optimum trade of type-C investors,

dNT;tþdt ¼NtþdtPtþdtð1� aÞ � NC;tPtþdtðb� aÞ � aWT;tþdtð1� aÞ � bWC;tþdtð1� bÞ

Ptþdt

hltþdt

�1dC� 1

dT

�þ 4C;tþdt

dC � 4T;tþdt

dT

i : (B3)

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